Parent Functions Calculator - Transformations, Tables and Graphs

Calculator Inputs

Use any parent family, apply transformations, evaluate a point, and generate a table and graph.

Parent function family

Vertical scale a

Horizontal scale b

Horizontal shift h

Vertical shift k

Evaluate at x

Table and graph x minimum

Table and graph x maximum

Step size

Base for exponential and logarithmic families

Angle unit for sine and cosine

Formula Used

The calculator applies every family through one transformation rule:

y = a · f(b(x − h)) + k

a changes vertical stretch and reflection.
b changes horizontal stretch, compression, and reflection.
h shifts the graph left or right.
k shifts the graph up or down.

Common parent rules

Linear: f(x) = x
Quadratic: f(x) = x²
Cubic: f(x) = x³
Absolute value: f(x) = |x|
Square root: f(x) = √x
Cube root: f(x) = ∛x

More parent rules

Reciprocal: f(x) = 1 / x
Exponential: f(x) = bˣ with b > 0 and b ≠ 1
Logarithmic: f(x) = log_b(x) with b > 0 and b ≠ 1
Sine: f(x) = sin(x)
Cosine: f(x) = cos(x)

How to Use This Calculator

Select the parent family you want to study.
Enter transformation values for a, b, h, and k.
Set a specific x-value for direct evaluation.
Choose the graph interval and table step size.
Use the base field for exponential or logarithmic families.
Pick degrees or radians for sine and cosine.
Press the calculate button to see the result above the form.
Review the equation, intercepts, table, graph, and export files.

Example Data Table

Family	Parent rule	Example transform	Example point	Output
Linear	f(x) = x	y = 2(x - 1) + 3	x = 4	9
Quadratic	f(x) = x²	y = (x - 2)² - 1	x = 5	8
Absolute value	f(x) = \|x\|	y = -3\|x + 1\| + 4	x = 2	-5
Exponential	f(x) = 2ˣ	y = 0.5 · 2^(x - 1) + 1	x = 3	3

FAQs

1. What is a parent function?

A parent function is the simplest graph in a family. Transformations build new functions from that original pattern without changing the basic family behavior.

2. What does a do in the formula?

The value a changes vertical size. Positive values keep orientation, negative values reflect across the x-axis, and larger magnitudes increase steepness or amplitude.

3. What does b do in the formula?

The value b changes horizontal behavior. Larger absolute values compress the graph, smaller nonzero values stretch it, and negative values reflect across the y-axis.

4. Why do some outputs show Undefined?

Undefined results appear when the chosen x-value violates the function domain. Common examples include square roots of negative inputs, logs of nonpositive inputs, or division by zero.

5. How are x-intercepts estimated here?

The calculator scans the sampled table and graph interval for sign changes. It then estimates roots numerically, so accuracy improves with a tighter step size.

6. Why do exponential and logarithmic families need a base?

Those families depend on a growth or log base. The base must be positive and not equal to one so the function stays mathematically valid.

7. Should I choose degrees or radians for trig functions?

Choose the unit that matches your course or given problem. The calculator converts degree inputs for sine and cosine before plotting and evaluating values.

8. What do the CSV and PDF buttons export?

The CSV export downloads the calculated x-y table. The PDF export saves the summary details and current result table for quick sharing or printing.